Q$3[17+3$ Marks$]$

A governor of Hartnell type, with the dimensions are shown in figure, runs at mean speed of $500rpm,$ each ball has a mass $3kg$ and $5\%$ reduction of speed causes a sleeve movement of $6mm.$

$($a$)$ If the ball arm is vertical at mean speed, find the stiffness in $\frac{N}{m}m.$ Take the arms obliquity into account. Neglect sleeve friction.

$($b$)$ Write the equation for this governor when it operates with the spindle being horizontal